Geometry Proofs

Polygon Int. Angles Thm.

The sum of the measures of the interior angles of a convex n-gon is (n-2) 180

Corollary to the Polygon Int. Angles Thm.

The sum of the interior angles of a quadrilateral is 360 degrees

Polygon Exterior Angles Thm.

The sum of the exterior angles of a polygon, one at each vertex, is 360

Parallelogram Opposite Sides Thm.

If a quadrilateral is a parallelogram, then it’s opposite sides are congruent

Parallelogram Opposite Angles Thm.

If a quadrilateral is a parallelogram, then it’s opposite angles are congruent

Parallelogram Consecutive Angles Thm.

If a quadrilateral is a parallelogram, then it’s consecutive angles are supplementary

Parallelogram Diagonals Thm.

If a quadrilateral is a parallelogram, then it’s diagonals bisect each other

Parallelogram Opposite Sides Converse

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram

Parallelogram Opposite Angles Converse

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram

Parallelogram Opposite Sides Parallel and Congruent Theorem

If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram

Parallelogram Diagonals Converse

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram

Rhombus Corollary

A quadrilateral is a rhombus if and only if it has 4 congruent sides

Rectangle Corollary

A quadrilateral is a rectangle if and only if it has 4 right angles

Square Corollary

A quadrilateral is a square if and only if it is a rhombus and a rectangle

Rhombus Diagonals Theorem

A parallelogram is a rhombus if and only if its diagonals are perpendicular

Rhombus Opposite Angles Theorem

A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles

Rectangle Diagonals Theorem

A parallelogram is a rectangle if and only if its diagonals are congruent

Isosceles Trapezoid Base Angles Theorem

The base angles of an isosceles trapezoid are congruent

Isosceles Trapezoid Base Angles converse

If a trapezoid had a pair of congruent base angles, then it is an isosceles trapezoid

Isosceles Trapezoid Diagonals Theorem

The diagonals of an isosceles trapezoid are congruent

Trapezoid Midsegment Theorem

The midsegment of a trapezoid is parallel to each base and its length is half the sum of the lengths of the basesT

Kite Diagonals Theorem

The diagonals of a kite are perpendicular to each other and one diagonal bisects the other.

Kite Opposite Angles Theorem

A kite has exactly one pair of congruent opposite angles

Corresponding Parts of Similar Polygons

If two triangles are similar then the ratios of the corresponding side lengths are equal to each other

Perimeters of Similar Polygons Thm.

Ratio of perimeters is equal to the ratio of their corresponding side lengths or k

Areas of Similar Polygons Thm.

If two polygons are similar then the ratio of the areas is equal to the square of their corresponding side lengths

Angle-Angle Similarity Theorem (AA)

If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar

Side-Side-Side Similarity Theorem (SSS)

If the corresponding side lengths of two triangles are proportional, then the triangles are similar

Side-Angle-Side Similarity Theorem (SAS)

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.\

Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally

Converse of the Triangle Proportionality Theorem

If a line divides two sides of a triangle proportionally, then it is parallel to the third side

Three Parallel Lines Theorem

If three parallel lines intersect two transversals, then they divide the transversals proportionally

Triangle Angle Bisector Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides

Pythagorean Theorem

In a right triangle, the sum of the squares of the lengths is equal to the square of the length of the hypotenuse (a² + b² = c²)

Converse of the Pythagorean Thm.

If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right angle

Pythagorean Inequalities Thm

For any triangle ABC, where c is the length of the longest side, the following statements are true: if ( c^2 < a^2 + b^2 ), then the triangle is acute; if ( c^2 > a^2 + b^2 ), then the triangle is obtuse.

45-45-90 Right Triangle

In a 45-45-90 triangle, both legs are congruent and the length of the hypotenuse is root 2 times the length of the legs

30-60-90 Right Triangle

In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is root 3 times the length of the shorter leg

Right Triangle Similarity Theorem

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles are similar to the original triangle and to each otherThis similarity holds true because corresponding angles are equal and the sides are proportional.

Geometric Mean (Altitude) Theorem

The length of the altitude to the hypotenuse of a right triangle is the Geometric Mean of the lengths of the segments of the hypotenuse

Geometric Mean (Leg) Theorem

The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the adjacent hypotenuse segment and the length of the hypotenuse

Area of an Oblique Triangle

A = ½ ab(sinC) or A = ½ bc(sinA) or A = ½ ac(sinB)

Law of Sines

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ .

Law of Cosines

a² = b² + c² - (2bc cosA) b² = a² + c² - (2ac cosB) c² = a² + b² - (2ab cosC)

Tangent Line to a Circle Theorem

In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of a circle at its endpoint on the circle

External Tangent Congruence Theorem

Tangent segments from a common external point are congruent

Arc Addition Postulate

Te measure of an arc formed by 2 adjacent arcs is the sum of the measure of the 2 arcs

Congruent Circles Theorem.

2 circles are congruent if and only if they have the same radius

Congruent Central Angles Theorem

In the same circle, or congruent circles, 2 minor arcs are congruent if and only if their corresponding central angles are congruent

Similar Circles Theorem

All circles are similar

Congruent Corresponding Chords Thm

In the same circle, or in congruent circles, 2 minor arcs are congruent if and only if their corresponding chords are congruent

Perpendicular Chord Bisector Thm.

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc

Perpendicular Chord Bisector Converse

If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameterand the two chords lie in the same circle.

Equidistant Chords Thm

In the same circle, or in congruent circles, 2 chords are congruent if and only if they are equidistant from the center

Measure of an Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of the intercepted arc

Inscribed Angles of a Circle Theorem

If 2 inscribed angles of a circle intercept the same arc, then the angles are congruent

Inscribed Right Triangle Theorem

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angleI

Inscribed Quadrilateral Theorem

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary

Tangent and Intersected Chord Theorem

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of the intercepted arc

Angles Inside the Circle Theorem

If 2 chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle

Angles Outside the Circle Theorem

In a tangent and a secant, 2 tangents, or 2 secants intersect outside a circle, then the measure of the angle formed is ½ the difference of the measures of the intercepted arcs.

Circumscribed Angle Theorem

The measure of a circumscribed angle is equal to 180 degrees minus the measure of the central angle that intercepts the same arc

Segments of Chords Theorem

If 2 chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments to the other chord

Segments of Secants Theoremng

If 2 secant segments share the same endpoint outside a circle, then the product of one secant segment and its external segment is equal to the same product of the other

Segments of Secants and Tangents Theorem

If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the secant segment and its external segment is equal to the square of the tangent segment.